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## Lemon Pledge and M&M's

According to Vince Staten's book Can You trust a Tomato in January? (1994)....

1. Lemon Pledge furniture polish contains more lemons than Country Time Lemonade.
2. M&M's are mixed in these color ratios: 3 brown, 2 yellow, 2 red, 1 green, 1 orange, and 1 tan.
Question: Find systematic ways to test the validity of these claims. Are they true?

D.C. (Bellingham) offers these approaches....
1. I would obtain some chemical process to determine the mixture ratio of the lemon juice versus the overall volume, and run several trials on each of the Country time lemonade and the furniture polish. The lemonade will be first strained for any pulp and the pulp will be weighed and incorporated into the total ratio. If necessary, different variations of the lemonade may be tested (pulp, no pulp...). After (for thoroughness) about 50 trials of determining the ratio, I would average and do a hypothesis test via regression on the furniture polish and each of the lemonade ratios to see if they are equal on average, or if the furniture polish, via a t-test, has a statistically significant greater size proportionally to the lemonade. I would then conclude to a certain confidence interval (likely 95%) whether the lemonade has less lemon in it than the polish.
2. I would buy several bags (lets go with 100 this time) and count out the number of each respective color in each bag, and record them in a table. I would then test their probablity ratios as described in the problem via a test on a multinomial distribution with colors assigned as p's (e.g. pbrown=3/10). I would use the test to see if there is a statistical difference in the probabilities estimated and their estimations. Based on this I would conclude at a certain level of confidence (likely 95%) whether the actual mixture differs in its ratio from the specified mixture.